Weakly nonlinear waves in stratified shear flows

In this paper we express the linearized dynamics of interacting interfacial waves in stratified shear flows in the compact form of action-angle Hamilton’s equations. The pseudo-energy serves as the Hamiltonian of the system, the action coordinates are the contribution of the interfacial waves to the wave action and the angles are the phases of the interfacial waves. The term ‘generalized action angle’ aims to emphasize that the action of each wave is generally time dependent and this allows for instability. An attempt is made to relate this formalism to the action at a distance resonance instability mechanism between counter-propagating vorticity waves via the global conservations of pseudo-energy and pseudo-momentum.

mKdV Equation for the Amplitude of Solitary Rossby Waves in Stratified Shear Flows with a Zonal Shear Flow

A two-timing perturbation technique is applied in order to investigate the mutual interaction between a Stokes wave train on the water surface and long, low-frequency, internal waves in a two-layer model. The set of equations derived conserves energy in the system of surface and internal waves. This conservation is attributable to the coupling terms that appear in both the internal-wave equations and the surface-wave equations and that describe the energy transfer between both wave families. In addition to the coupling and nonlinear effects included in the surface-wave equations (which are part of the interaction equations), a higher order, linear effect, attributable to the slow variations in surface-wave properties, is also included. Because of the availability of a rigorously ordered description of the interaction processes, it has been possible to assess the relative influence of the nonlinear, finite-amplitude terms, the coupling terms, and the higher order linear terms in the complete set of governing equations for a given set of conditions.

<p>The interaction of small-scale turbulence with internal and surface waves is an urgent problem of hydrology and oceanology. In particular, this issue is especially important for the properties of the upper layer of the ocean and the inland waters.</p><p>Small-scale processes that exist against the background of average profiles of various hydrophysical quantities (temperature, velocity, density, and large-scale currents caused, in particular, by wind forcing) are usually nonlinear and therefore effectively interact with each other. We consider some aspects of the interaction of internal waves and turbulence in the upper layer of the ocean and inland waters within the framework of the semi-empirical theory of turbulence in a stratified fluid. The model used in this study takes into account  mutual transformation of the kinetic and potential energies of turbulent fluctuations [Ostrovsky&Troitskaya, 1987; Zilitinkevich et al., 2013]. The effects of amplification and maintenance of turbulence by low-frequency and high-frequency internal waves, quasi-stationary distributions of turbulent energy in the presence of a shear caused by a low-frequency internal wave are investigated; the role of the transformation of energies on the indicated processes is analyzed.</p><p>A modification of the k-epsilon mixing scheme is also proposed, which removes the limitation on the existence of turbulence at large values of the gradient Richardson number. Within the framework of the modification, the parameterization of the Prandtl number is used, which makes it possible to take into account the influence of density stratification and velocity shear on mixing processes.</p><p>A numerical study of the influence of vertical mixing schemes on the transfer processes of biochemical fields in an internal reservoir was also carried out. The modified scheme was implemented into a three-dimensional model of thermo-hydrodynamics and biochemistry of an inland water body [Gladskikh et al., 2021], and a series of numerical experiments was conducted.</p><p>The work was supported by the RFBR (20-05-00776; 20-05-00322; 21-05-52005), and by Moscow Center of Fundamental and Applied Mathematics (agreement with the Ministry of Science and Higher Education 075-15-2019-1621).</p><p>Ostrovsky LA, Troitskaya YuI (1987) A model of turbulent transfer and dynamics of turbulence in a stratified shear flow. Izv Akad Nauk SSSR, Fiz Atmos Okeana. 3:101–104.<br>Zilitinkevich SS, Elperin T, Kleeorin N, Rogachevskii I, Esau I (2013) A hierarchy of Energyand Flux-Budget (EFB) turbulence closure models for stably-stratified geophysical flow. Boundary-Layer Meteorol. 146:341–373<br>Gladskikh DS, Stepanenko VM, Mortikov EV (2021) The Effect of the Horizontal Dimensions of Inland Water Bodies on the Thickness of the Upper Mixed Layer. Water Resour 48:226–234</p>

Abstract. A higher-order extension of the familiar Korteweg-de Vries equation is derived for internal solitary waves in a density- and current-stratified shear flow with a free surface. All coefficients of this extended Korteweg-de Vries equation are expressed in terms of integrals of the modal function for the linear long-wave theory. An illustrative example of a two-layer shear flow is considered, for which we discuss the parameter dependence of the coefficients in the extended Korteweg-de Vries equation.

The nonhydrostatic pressure effects on the generation and propagation of wind-forced internal waves are studied with a two-dimensional numerical ocean model. A one-way directed wind pulse over a stratified ocean initiates surface and internal waves in a closed basin. The studies are performed with horizontal grid sizes in the range from 1 km to 62.5 m. The experiments are performed with both a hydrostatic and a nonhydrostatic model, facilitating systematic studies of the sensitivity of the numerical model results to the grid size and to the nonhydrostatic pressure adjustments. The results show that the nonhydrostatic pressure effects are highly dependent on the grid size and grow with increased resolution. In the internal depression wave, the horizontal nonhydrostatic pressure gradients reach the same order of magnitude as the hydrostatic gradients in the high-resolution nonhydrostatic studies. In these studies, the nonhydrostatic pressure gradients approximately balance the corresponding hydrostatic pressure gradients in the internal depression wave, and the wave degenerates into a train of soliton waves. The time for the soliton form to develop agrees with the steepening timescale calculated from Korteweg-de Vries theory. In the high-resolution hydrostatic model, the internal depression wave takes the form of a single wave front. When the internal waves are generated in the boundary layers, the nonhydrostatic pressure gradients are much smaller than the hydrostatic gradients and the generation processes are not effected by the nonhydrostatic pressure with the present range of grid sizes.

The mesoscale eddy and internal wave both are phenomena commonly observed in oceans. It is aimed to investigate how the presence of a mesoscale eddy in the ocean affects wave form deformation of the internal solitary wave propagation. An ocean eddy is produced by a quasi-geostrophic model in f-plane, and the one-dimensional nonlinear variable-coefficient extended Korteweg-de Vries (eKdV) equation is used to simulate an internal solitary wave passing through the mesoscale eddy field. The results suggest that the mode structures of the linear internal wave are modified due to the presence of the mesoscale eddy field. A cyclonic eddy and an anticyclonic eddy have different influences on the background environment of the internal solitary wave propagation. The existence of a mesoscale eddy field has almost no prominent impact on the propagation of a smallamplitude internal solitary wave only based on the first mode vertical structure, but the mesoscale eddy background field exerts a considerable influence on the solitary wave propagation if considering high-mode vertical structures. Furthermore, whether an internal solitary wave first passes through anticyclonic eddy or cyclonic eddy, the deformation of wave profiles is different. Many observations of solitary internal waves in the real oceans suggest the formation of the waves. Apart from topography effect, it is shown that the mesoscale eddy background field is also a considerable factor which influences the internal solitary wave propagation and deformation.

A mechanism of atmospheric ozone disturbance by internal gravity wave in a stratified shear flow

Oscillating wave packets (breathers) are a significant part of the dynamics of internal gravity waves in a stratified ocean. The formation of these waves can be provoked, in particular, by the decay of long internal tidal waves. Breather interactions can significantly change the dynamics of the wave fields. In the present study, a series of numerical experiments on the interaction of breathers in the frameworks of the etalon equation of internal waves—the modified Korteweg–de Vries equation (mKdV)—were conducted. Wave field extrema, spectra, and statistical moments up to the fourth order were calculated.

Internal waves play a significant role in the resuspension and transport of fine sediment adjacent to the sea bottom in the coastal region. Inall (2009) found the horizontal current and diffusion of mass due to the breaking of internal waves in the Fjord by using a fluorescent tracer. Internal waves have been shown to cause crucial mass transport and affect the ecological system. In particular, an internal solitary wave that progresses without changing the profile contributes to mass transport on a sloping bottom. Therefore, it is essential to clarify how an internal solitary wave breaks over a slope and transports mass. When pycnocline thickness is negligible in a twolayer fluid, an internal solitary wave breaking over a slope can be categorized into four breaker types by applying the latest classification based on wave slope, bottom slope gradient and an internal Reynolds number. Aghsaee et al. (2010) demonstrated that an internal solitary wave breaking over a slope can be categorized into four breaker types: surging, collapsing, plunging, and fission breakers using three-dimensional numerical simulations. They used wave slope and bottom slope gradient. However, some plunging and collapsing breakers were not appropriately categorized. Nakayama et al. (2019) solved the classification problem by introducing an internal Reynolds number based on the Korteweg–De Vries equation. However, it remains unsolved if this classification can categorize the breaking of an internal solitary wave under thick pycnocline conditions. This study uses numerical simulations to investigate the applicability of the classification under changing pycnocline thickness (Nakayama et al., 2021).

Detection of solitary ocean internal waves from SAR images by using U-Net and KDV solver technique

The long time effect of weak rotation on an internal solitary wave is the decay into inertia-gravity waves and the eventual formation of a localised wavepacket. Here this initial value problem is considered within the context of the Ostrovsky, or the rotation-modified Korteweg-de Vries (KdV), equation and a numerical method for obtaining accurate wavepacket solutions is presented. The flow evolutions are described in the regimes of relatively-strong and relatively-weak rotational effects. When rotational effects are relatively strong a second-order soliton solution of the nonlinear Schrödinger equation accurately predicts the shape, and phase and group velocities of the numerically determined wavepackets. It is suggested that these solitons may form from a local Benjamin-Feir instability in the inertia-gravity wave-train radiated when a KdV solitary wave rapidly adjusts to the presence of strong rotation. When rotational effects are relatively weak the initial KdV solitary wave remains coherent longer, decaying only slowly due to weak radiation and modulational instability is no longer relevant. Wavepacket solutions in this regime appear to consist of a modulated KdV soliton wavetrain propagating on a slowly varying background of finite extent.

The manifestation of internal wave (IW) in synthetic aperture radar (SAR) ocean images has always been of considerable interest to oceanographers. In this paper, the location of IW is carried out by the edge extraction algorithm based on wavelet transform modulus maxima, which shows the moment IW distribution on time and space. The propagation direction of IW is extracted by the fourier analysis and the division processing. The IW wavelength is extracted from the wavelet transform technology for the non-periodicity of IW signal. Based on the former research about the relation between the relative normalized radar cross section (NRCS) variations and the IW parameters, we adopt a iteration method to inverse some IW parameters including the mixed layer depth, amplitude and the propagation speed. From the results, the location image can basically catch IW information except a too weak IW signal. The inversed wavelength has minor larger values in the light wave rupture of the SAR image but it is consistent to the SAR image. The IW direction, mixed layer depth and propagation speed present a uniform distribution for the value in each position and show that these parameter are basically invariable for a IW packet at least. But the IW amplitude in different points varies intensely deviated from the mean, which means that it is vulnerable for the sea bottom topography and hydrodynamic modulation.

Internal solitary waves (ISWs) often seriously threaten the survivability of the underwater submersible. Most of the existing investigations focus on the interaction between ISWs and fixed or suspended submersibles. However, the investigation on the interaction between ISWs and self-propelled submersibles is still scarce, which is a more realistic case in the marine engineering. In this paper, a three-dimensional numerical model for the interaction between ISW and self-propelled submersible is developed. Based on the extended Korteweg–de Vries (eKdV) theory, the ISW is generated in a two-layer fluid numerical wave tank by solving the Navier–Stokes (N–S) equation. By introducing the hydrodynamic loadings provided by the ISW environment into the standard operation equation of the submersible, the six degrees-of-freedom motions of the submersible can be obtained. The dynamic overset mesh technology is adopted to simulate the motions of the self-propelled submersible in the ISW fluid field. The present numerical model is validated by comparing with the experiment on a submerged cylinder in the ISW environment. Using this numerical model, we compare the interaction effects of the fixed, suspended, or the self-propelled submersible in ISW and discuss the influences of propulsive forces of the self-propelled submersible on ISW–structure interaction effects. The numerical results show that the loadings and movements of the submersible change remarkably in the surge, heave, and pitch direction. Especially, the submersible with high propulsive forces can pierce the wave surface and reach a large pitch angle with the amplitude of 36°, which further results in a 35% motion speed decrease in the initial propulsion direction.

Observations on the lee of a topographic ridge show that the turbulence kinetic energy (TKE) dissipation rate due to shear instabilities is three orders of magnitude higher than the typical value in the open ocean. Laboratory-scale studies at low Reynolds number suggest that high turbulent dissipation occurs primarily within the core region of shear instabilities. However, field-scale studies indicate that high turbulence is mainly populated along the braids of shear instabilities. In this study, a high-resolution, resolving the Ozmidov-scale, non-hydrostatic model with Large Eddy Simulation (LES) turbulent closure is applied to investigate dominant mechanisms that control the spatial and temporal scales of shear instabilities and resulting mixing in stratified shear flow at high Reynolds number. The simulated density variance dissipation rate is elevated in the cusp-like bands of shear instabilities with a specific period, consistent with the acoustic backscatter taken by shipboard echo sounder. The vertical length scale of each cusp-like band is nearly half of the vertical length scale of the internal lee wave. However, it is consistent with instabilities originating from a shear layer based on linear stability theory. The model results indicate that the length scale and/or the period of shear instabilities are the key parameters to the mixing enhancement that increases with lateral Froude number FrL, i.e. stronger shear and/or steeper ridge.